Download THE DEFECT EFFECT ON THE ELECTRONIC CONDUCTANCE IN H. S. Ashour

THE DEFECT EFFECT ON THE ELECTRONIC CONDUCTANCE IN
BINOMIALLY TAILORED QUANTUM WIRE
H. S. Ashoura, A. I. Ass’adb, M. M. Shabatc
a
Department of Physics, Al-Azhar University of Gaza, Gaza Strip, P.A.
E-mail: [email protected]
b
Department of Physics, Al-Aqsa University, Gaza Strip, P.A.
E-mail: [email protected]
c
Department of Physics, Islamic University of Gaza, Gaza Strip, P.A.
E-mail: [email protected]
ABSTRACT
The paper considers the effect of the defects on the
electronic transmission properties in binomially tailored
waveguide quantum wires, in which each Dirac delta
function potential strength have been weight on the
binomial distribution law. We have assumed that a single
free-electron channel is incident on the structure and the
scattering of electrons is solely from the geometric nature
of the problem. We have used the transfer matrix method
to study the electron transmission. We found this novel
structure has a good defect tolerance. We found the
structure tolerate up to ±20% in strength defect and ±5%
in position defect for the central Dirac delta function in
the binomial distribution. Also, we found this structure
can tolerate both defect up to ±20% in strength and ±5%
in position dislocation.
1. INTRODUCTION
In the last decade, there was a growing interest in the
electron conductance through one-dimensional scattering
problems, especially in those cases where the potential is
periodic structure with finite number of identical cells [1,
2]. Because of the remarkable advances in nanotechnology and micro fabrication, it is possible to confine
electrons in a conductor with a lateral extent of 100nm or
less, resulting narrow quantum wire [3]. In these
mesoscopic devices, the electron transport is best
described by quantum mechanics. Miniature size of these
devices eliminates the defect of scattering. At a low
enough temperature, the motion of electrons through
these devices is ballistic or quasiballistic and the electronphonon interaction can be neglected. So that, the phase
coherence length enlarges enough when compared with
the device dimension. Mesoscopic devices can be
considered as a coherent elastic scatterer [3]. Therefore,
the electron transport properties, solely depends upon the
geometrical structure of the quantum waveguide.
In recent years, there has been a growing interest in the
electron transport through a sequence of Dirac delta
function potential [4-9]. The researcher used different
methods to study the electron transport in a waveguide
quantum wire [10-12]. Recently, Ashour et al [13] has
proposed a novel structure which is the binomially
tailored waveguide quantum wires, in which each Dirac
Delta function potential strength has been weight on the
binomial distribution law. In this paper, we study the
defects effect on the electronic conductance on the novel
structure proposed by [13].
In section 2, we outline the transfer matrix method which
connects the solutions at the ends of the waveguide
quantum wires. We introduce the novel structure of the
binomially tailored waveguide quantum wires. In section
3, we explore defect effects on the electronic conductance
spectrum through the binomially tailored quantum wire.
In this section, we have studied strength defect and
dislocation defects on the central Dirac delta function in
the distribution. Section 4 has been devoted to the
conclusions of this study.
2. TRANSMISSION MATRIX THOUGH
2.1 Periodic Structure
In this context, we consider a finite periodic structure of
Dirac delta function potential (Dirac Comb). Also, we
assumed that the structure is narrow enough so that just
single channel of electrons can be considered. In this
treatment, we neglected electron-electron interaction, and
we assume the temperature low enough so that electronphonon interaction can be neglected as well. We assumed
the scattering of electrons mainly form the geometrical
structure of the potential. The potential can be written as
follows:
V ( x ) = ∑ U jδ ( x − x j )
N
(1)
j =1
Thus, U j and x j represent the strength and the position
of the j th delta function respectively, and N is the
number of the Dirac delta functions in Dirac Comb. The
distance between the adjacent barriers are given by
d j = x j +1 − x j . The Schrödinger wave equation of one
dimension can be written as follows:
−
2
h2 d ψ ( x )
+ V ( x )ψ ( x ) = Eψ ( x )
2m* dx 2
(2)
Thus, M t ( 2,2 ) is the second element in the second row
in a 2 × 2 matrix. According to the Landauer-Buttiker
formula, the electron conductance through this structure
is [19,20]
G=
−1
(7)
parameters, Ω j = md jU j / π h . In figure (1), we show
2
2
the conductance through N = 5 Dirac delta function
potential with strength Ω = 0.2 . A perfect transmission
in this case is in general impossible as predicted by [4,
22]. According to reference [21] we can not have a
resonant transmission, T = 1 , even if N is very large.
1.20
*
ikx
ikx
⎛C⎞ ⎛e j 0 ⎞ ⎛1−iβj −iβj ⎞⎛e j 0 ⎞⎛ A⎞
(3)
⎟ ⎜
⎟
⎟⎜
⎜ ⎟ =⎜⎜
−ikx
−ikx ⎜ ⎟
⎝ D⎠ ⎝ 0 e j ⎟⎠ ⎝ +iβj 1+iβj ⎠⎜⎝ 0 e j ⎟⎠⎝ B⎠
2
We assume a dimensionless strength for Delta function
potential
[21]
by
rescaling
our
Thus, V ( x ) is the periodic potential given by equation
(1), m is the electron effective mass, which is
considered approximately constant over the interaction
range. The solution of Schrödinger wave equation for
single Delta function potential can be found in the
literature and also in the transfer matrix formulism [1416, 17]. The transfer matrix for periodic structure has
been used also to study the transmission of electron
through Comb structure [4-6, 14-16, 17]. The transfer
matrix, which is related to the input electron wave and the
output, is given by [4-6]
2e 2
T
h
1.00
0.80
G
0.60
0.40
0.20
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
βj
is γ
j
/ 2k , where γ j is 2m* U j / h 2 , and k
*
2
is the wave number given by 2m E / h . So that the
transfer matrix at a given single barrier can be written as,
M j = S −1 ( ikx j ) ⋅ Γ ( β j ) ⋅ S ( ikx j )
(4)
The total transfer matrix which represents the electron
propagating through the entire device is just the repetitive
product of the transfer matrix of a single barrier. We find
⎡ 1
⎤
Mt = S−1 ( ikdN ) ⋅Γ( βN ) ⋅ ⎢ ∏ S ( ikd j ) ⋅Γ( β j )⎥ ⋅ S ( 0) (5)
⎣ j =N−1
⎦
Thus, d j is the periodic spacing between two adjacent
Dirac Delta functions.
Then the transmission amplitude is given by [18],
T=
1
M t ( 2, 2 )
(6)
5.00
kd / π
Figure 1: Conductance spectrum G in the units of
Thus,
4.50
2e 2 / h as a
function of kd / π for a sequence of Dirac delta function
potential with N=5. The strength of the potential here is
Ω = 0.2 .
2.2 The Binomially Tailored Quantum Wire (BTQW)
In this subsection, we reintroduce BTQW structure as
shown in figure 2. The Dirac delta function has been
equally spaced but their strength, Ω j , has been
weighted according to the binomial distribution law,
which is
⎛N⎞
Ω ( N j ) = ⎜ ⎟ / 2 N , N j = 0,1,...., N
⎝Nj ⎠
(8)
1.20
1.00
Ω( N j )
0.80
G
0.60
0.40
Nj
0.20
Figure 2: Shows binomially tailored Dirac delta function
potential.
N j values weighted by
0.00
0.00
equation (8).
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
kd / π
Thus, Ω ( N j ) represents the strength of the Dirac delta
Figure 3: Conductance spectrum G in the units of 2e 2 / h as a
function of kd / π for a BTQW, with N=5.
potential, N + 1 represents the total number of Dirac delta
function potentials in the quantum wire, and N j is the
order of the Dirac delta potential. This novel structure of
quantum wires can be released by putting metallic gates
on top of a one dimensional electron gas and then by
applying voltages, according to the binomial distribution
law, to deplete the electron gas underneath. In this case,
equation (9) is no longer valid for our new structure. So
that the total transmission matrix can be written as
follows:
M t = S −1 (ikd N +1 ) ⋅ Γ(β N +1 ) ⋅ S (ikd N ) ⋅ Γ(β N ) ×
........... × S (ikd1 ) ⋅ Γ(β1 ) ⋅ S (0)
(9)
Notice that the potential strength is weighted according to
equation (8). In figure 3, we show the conductance
spectrum through a sequence of a binomially tailored
Dirac delta function potentials. It is quite interesting to
notice that we have reached a transmission through this
structure approaches to unity in the allowed band region
without any ripples after some k value. Here, we have a
resonant tunneling due to coherent interference effects
due to elastic scattering of electrons, which leads the
transmission to reach unity and also to have constant
value over the allowed band or conduction band. Also,
we see that there is a forbidden band or conduction gap
where the transmission is small.
3. DEFECTS EFFECT ON THE ELECTRONIC
CONDUCTANCE OF BTQW
3.1 Strength Defect
In this subsection, we study effect of strength defect of
the central element of the binomial tailored quantum
wire, and keeping the other elements and the spacing
between the Dirac delta function potentials constant, on
the electronic conductance through the BTQW. First, we
consider the strength defect does not exceed ±5% of the
Dirac delta function potential strength. That is, when the
central Dirac delta functions potentials strength
is Ω j ( N / 2 + 1) ± 0.05Ω j ( N / 2 + 1) . In figure 4-a, we plot the
electronic conductance spectrum for both strengths with
N j is 35 and scaling factor of three. As can noticed there
is slight difference between the two conductance
spectrum curves, and defect free curves. In figure 4-b, we
have increased the strength defect up to ±20% , we have
noticed some measurable differences between the two the
conductance spectrum curves and defect free curves, but
still the conduction band and the forbidden bands well
defined, which is a very good feature for BTQW.
between the central Dirac delta function and the adjacent
one. In figure 5-b, we increase the dislocation defect up
to ±20% , we have noticed measurable differences
between the two curves and that of no defect case, but
still the conduction band is well defined but the forbidden
bands have a split compared to forbidden band in no
defect curves. This splitting is due to resonant state in the
forbidden energy band which leads to a bound state in the
structure [3]. This is because the particle mode cannot
propagate and hence get trapped.
1.20
1.00
0.80
0.60
G
0.40
1.20
0.20
1.00
0.00
0.00
0.50
1.00
1.50
2.00
2.50
kd / π
3.00
3.50
4.00
4.50
0.80
5.00
Figure 4-a: The electronic conductance, in the units of 2e 2 / h as
a function of kd / π . In this case, the defect is only ±5% , in the
strength of the central Dirac delta function.
G
0.60
0.40
1.20
0.20
1.00
0.00
0.00
0.50
1.00
1.50
2.00
0.80
G
2.50
3.00
3.50
4.00
4.50
5.00
kd / π
Figure5-a: The electronic conductance, in the units of 2e 2 / h as
a function of kd / π . In this case, the defect is only ±5% , in the
position of the central Dirac delta function.
0.60
1.20
0.40
1.00
0.20
0.80
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
kd / π
Figure 4-b: The electronic conductance, in the units of 2e 2 / h as
a function of kd / π . In this case, the defect is only ±20% , in
the strength of the central Dirac delta function.
G
0.60
0.40
0.20
3.2 Dislocation Effect
In this subsection, we study dislocation defect effect on
the position of the central element in the BTQW, and
keeping the other elements and the spacing between the
Dirac delta function potentials constant. First, we
consider the position defect does not exceed ±5% of the
Dirac delta function potential spacing constant. That is,
when the central Dirac delta function potentials spacing is
d ± 0.05d . In figure 5-a, we plot the electronic
conductance spectrum for both dislocations with
N j is 35 and scaling factor of three. Compared to defect
curves, as can noticed there is a difference between the
two curves. The conduction band starts lose its flatness
and the forbidden band shaper for increased spacing
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
kd / π
Figure 5-b: The electronic conductance, in the units of 2e 2 / h as
a function of kd / π . In this case, the defect is only ±20% , in
the position of the central Dirac delta function.
4. CONCLUSION
We found the novel structure introduced by [13] has a
good tolerance for strength reaches up to ±20% and
dislocation tolerance reaches up to ±5% without losing
the fascinating electronic transmission characteristics.
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